Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $k = \dfrac{x}{4(4x + 9)} \div \dfrac{-8}{3(4x + 9)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{x}{4(4x + 9)} \times \dfrac{3(4x + 9)}{-8} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ x \times 3(4x + 9) } { 4(4x + 9) \times -8 } $ $ k = \dfrac{3x(4x + 9)}{-32(4x + 9)} $ We can cancel the $4x + 9$ so long as $4x + 9 \neq 0$ Therefore $x \neq -\dfrac{9}{4}$ $k = \dfrac{3x \cancel{(4x + 9})}{-32 \cancel{(4x + 9)}} = -\dfrac{3x}{32} = -\dfrac{3x}{32} $